# Random walk

## Introduction

A *random walk* is a stochastic process which describes a path made of consecutive random steps.

## Gaussian

In Gaussian random walk the steps follow a continous Gaussian distribution. We will look at two different types, the univariate and multivariate kind.

### Univariate

A univariate Gaussian Random Walk, is a series of *i.i.d.* \(\mathcal{N}(0,1)\) random variables such that

\begin{align*} X_0&=0 \\ X_t&=X_{t−1}+\epsilon_t \end{align*}

Where \(t=1,2,\dots\) and \(\epsilon_t\) is a series of *i.i.d.* \(\mathcal{N}(0,1)\) random variables.

Let’s illustrate a simple univariate Gaussian random walk in Python, by plotting 1000 realisations.

```
import numpy as np
N = 1000
np.random.seed(23)
realisations = []
for i in range(N):
realisations.append(np.cumsum(np.random.normal(size=100)))
```